Applied Physics Research; Vol. 7, No. 5; 2015 ISSN E-ISSN Published by Canadian Center of Science and Education

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1 Appled Pyss Resear; Vol. 7, No. 5; 2015 ISSN E-ISSN Publsed by Canadan Center of Sene and Eduaton oward te rue Seond Law Part II: e Untenable Assumpton of Current Seond Law ermodynams and te Come nto beng of te Negentrop Formulaton José C. Íñguez 1 1 Retred nstrutor of sene and matemats, st Street, Douglas AZ 85607, USA Correspondene: José C. Íñguez, Retred nstrutor of sene and matemats, st Street, Douglas AZ 85607, USA. el: E-mal: nguez.jose@gmal.om Reeved: July 30, 2014 Aepted: August 7, 2015 Onlne Publsed: August 12, 2015 do: /apr.v7n5p39 URL: ttp://dx.do.org/ /apr.v7n5p39 Abstrat Seond law termodynams, as urrently understood, s ere proved to be a body of knowledge essentally dfferent from tat of Clausus. e rejeton on logal grounds of ts bas tenet eat and work as energy forms of te same qualty n ersble proesses- leads, troug a son of Clausus work, to a new body of knowledge, te Negentrop Formulaton, taken by ts autor to be te true seond law of termodynams. In t te total entropy ange for any gven work-produng termodynam proess s found to be determned by te ombnaton of te opposte sgn ontrbutons of te entrop (work degradng) and negentrop (work produng) transformatons n t takng plae. e effeny-dependent palene of one of tese oppostes over te oter opens te door for rersble proesses wt postve, negatve, or zero total entropy anges. s noton s at te enter of a testable predton pertanng self-organzng penomena. Keywords: Clausus termodynams, te seond law, te transformaton of eat nto work, te negentrop formulaton, entropy, negentropy 3. Current Seond Law ermodynams Fnds Orgn n te Correton of Clausus Work 3.1 Introduton Most students and studous of termodynams beleve tat seond law termodynams as t s known, wrtten about, and taugt nowadays s essentally Clausus work. Small anges ere and tere, most lkely; a lot extended, for ertan; but all n all te termodynams Clausus onstruted by orretng and extendng Carnot s work on eat engnes and te motve power of eat. s belef, owever, bears no orrespondene wt realty. e verson of seond law termodynams tat te present autor and you te reader as been exposed to s atually a radally dfferent verson from te body of knowledge onsttutng Clausus work on ts matter. s body of knowledge w wll be ere referred to as urrent seond law termodynams or C- must ave emerged from an effort to orret te logal sortomngs of Clausus work; tose exposed n te pous paper of ts two part seres. It seems reasonable to tnk tat wen tese flaws beame known, te law of nreasng entropy, apart from beomng te bakbone te essental noton- of seond law termodynams, as well as an mportant onept n plosopy and eonoms, among oter parels of knowledge, ad already aqured te status of te supreme law of nature (Eddngton, 1929, p.74), and as su taken to be beyond doubt and rtsm. In ts dogmat atmospere t s not unreasonable to tnk tat te orretons of te sad flaws were onduted n a way su tat te rule of te law of nreasng entropy remaned supreme, undsputed. As to te wo and te wen of tese orretons, te present autor as no fatual knowledge. Wat s ndeed fatual, as wll be proved below, s tat te aevement of ts objetve demanded te replaement of te non-zero values for te two transformatons takng plae n a ersble ylal proess -notons upon w Clausus onstruted s verson of te seond law of termodynams- for values of magntude equal to zero. e unexplaned, surrepttous, and seretve nature of te replaement puts t, n ts autor s opnon, n a par wt an at of mag, or, f te oxymoron s allowed, of magal sene. It appears tat no effort was made to ntrodue te neessary orretons nto Clausus work n order to brng to fruton te goal e unsuessfully 39

2 Appled Pyss Resear Vol. 7, No. 5; 2015 pursued: tat of brngng forward a law w refleted nature s beavor n regard to eat-work nter-onversons, even f te results were n opposton to te putatve supreme law of nature. In arbtrarly modfyng Clausus values to retan or guarantee te valdty of te law of nreasng entropy wat we got n return was a dstorted mage of realty, or to be more prese, an mage tat only under ertan ondtons orresponds to tat of nature; or even peraps a onstruton to sut te purposes of an deology, not an mplausble possblty wen due onsderaton s gven to te fat tat, as te followng quote from Bazarov llustrates, te law of nreasng entropy ad beome a sort of fnal argument for or aganst onfltng plosopal postons (See note 1): e (seond) law as augt te attenton of poets and plosopers and as been alled te greatest aevement of te nneteent entury. Engels dslked t, for t supported opposton to Daletal Materalsm, wle Pope Pus XII regarded t as provng te exstene of a ger beng. (Bazarov, 1964) e restoraton of self-onssteny and order produed by tese anges was, owever, only apparent, as tey were aeved va te replaement of one flaw for anoter. at ts s ndeed te ase wll be ere proved by sowng tat te angular stone of urrent seond law termodynams s te absurd noton assertng tat n ertan stuatons eat and work are, and n some oters are not, energy forms of dfferent qualty. ese matters are dealt wt n Setons 3.2 and 3.3 below. e numberng of setons, equatons, fgures and tables ere used ontnues from tose of part I. 3.2 Current Seond Law ermodynams Emerges from te Correton of Clausus Work e unvelng of te essental dfferene exstng between Clausus formulaton and C starts by takng anoter look at tat smple proess known as te sotermal and ersble expanson of an deal gas. s proess, orrespondng for te purposes of ts dsusson to proess AE n Fgure 2(b) and represented n detal n Fgure 3 of Part I, s, t sould be remembered, te one responsble for brngng forward te work output of te yle n te form of transformaton [ ( ) W]. It s to te determnaton of te entropy ange assgned by C to ts transformaton tat te followng argument s dreted to. Irrespetve of wat perspetve Clausus or C- s adopted to study te sotermal and ersble expanson, one fat remans true, before any amount of work dw an be delvered to te meanal reservor, te equvalent amount of eat d as to flow from te eat reservor to te gas. s noton leads, as dsussed n Seton 2.9, to te realzaton tat two transformatons are takng plae n su a proess: te transformaton of eat between two bodes te reservor and te gas- of essentally te same temperature, and te transformaton of ts eat nto work by te opposed expanson of te gas. In referene to te fnte amounts of eat and work and W te pous notons lead to te followng expresson for te total entropy ange of ersble proess AE: D S [ AE] = D S [ ( ) ( )] +D S [ ( ) W] (38) C C C In te pous equaton te sub-ndex C stands for urrent seond law termodynams. For easy omparson wt te results of part I we ave desgnated te proess under onsderaton as AE, w s te notaton orrespondng to Fgure 2(b). e fat tat te entropy ange assoated to te transfer of eat between two bodes of essentally te same temperature s -also wt ndependene of te perspetve adopted- equal to zero, allows wrtng Equaton (38) as follows D S [ AE] = D S [ ( ) W] (39) C C e substtuton n Equaton (39) of te zero total entropy ange assgned by C to an sotermal and ersble expanson, or to any oter ersble proess for tat matter (Ptzer & Brewer, 1961, p. 83),.e. D S [ AE] = 0 (40) leads us to te dentfaton of te entropy ange assgned by C to transformaton [ ( ) W] : C D S [ ( ) W] = 0 (41) C e total entropy ange for sotermal and ersble ompresson EA, te nverse of te one just dsussed, an now be wrtten as follows D S [ EA] = D S [ ( ) ( )] +D S [ W ( )] (42) C C C e same argument leadng from Equaton (38) to (39) an be used to transt from Equaton (42) to te followng expresson for te total entropy ange of proess EA: D S [ EA] = D S [ W ( )] (43) C C 40

3 Appled Pyss Resear Vol. 7, No. 5; 2015 Invokng agan te zero total entropy ange assgned by urrent seond law termodynams to any ersble proess ( D SC [ EA] = 0) produes te followng entropy ange for [ W ( )] : D S [ W ( )] = 0 (44) C Armed wt ts knowledge we an now proeed to evaluate te entropy ange for te ersble transformaton of eat as t ours n a ersble eat engne. As noted n Seton 2.4 of Part I, ts transfer of eat s te sole produt of te onatenaton of proesses EB-BC-CD-DA. A representaton of ts onatenaton s gven n Fgure 6, below. If so, te followng expresson an be wrtten D S [ ( ) ( )] =D S [ EB] +D S [ BC] +D S [ CD] +D S [ DA] (45) C C C C C e already noted sentrop nature of ersble proesses BC and DA an be expressed n te followng manner e ombnaton of Equatons (45) and (46) leads to D SBC [ ] +D SDA [ ] = 0 (46) D S [ ( ) ( )] = D S [ EB] +D S [ CD] (47) C C C e fat tat sotermal and ersble proesses EB and CD are, respetvely, of te same nature as proesses AE and EA, allows us to wrte ter respetve total entropy anges after Equatons (39) and (43) as follows: D S [ ( ) ( )] = D S [ ( ) W ] +D S [ W ( )] (48) C C C e fat made evdent by Equatons (41) and (44) tat for urrent seond law termodynams te ersble transformatons of eat nto work and ve-versa take plae wt a zero entropy ange leads us, fnally, to te followng total entropy ange for te onatenaton of proesses EB-BC-CD-DA or equvalently, for transformaton [ ( ) ( )],.e., For ts nverse we an wrte D S [ ( ) ( )] = 0 (49) C D S [ ( ) ( )] = 0 (50) C We an now proeed to alulate te total or unverse entropy ange for a ersble eat engne n terms of te ombned total entropy anges for proesses AE and EB-BC-CD-DA, or equvalently, as allowed by Equatons (39) and (45), n terms of te ombned entropy anges of te two transformatons brougt forward by tose proesses, as follows D S [ Cyle] = D S [ ( ) W] +D S[ ( ) ( )] = 0+ 0= 0 (51) C C Note tat te same total entropy ange for a ersble eat engne s provded by bot, Clausus onstruton and urrent seond law termodynams. e pat leadng to ts result s, owever, radally dfferent n one and te oter. Clausus produes a zero, as Equaton (29) makes evdent, va te addton of two non-zero quanttes. Current termodynams does t, on ts part as te pous equaton sows- va te addton of two zeros. In order to be able to fully ontrast Clausus formulaton wt C we must develop for te latter a table smlar to able 1. In order to do ts we must asertan te values C asrbes to te followng transformatons: [ ( ) ( )] rr and [ W ( )] rr. e former an be obtaned va te applaton of Equaton (13) to te eat reservors takng part n ts rersble eat transfer proess. Wen ts s done we get: D SC [ ( ) ( )] = - rr + (52) As for [ W ( )] rr all we ave to do s to arry over from noted C textbook autor s Ptzer and Brewer (1961, pp ) and Smdt (1966, p. 130) te value by tem asrbed to te rersble degradaton of work nto eat: D S [ W ( )] = / (53) C evdent n ter followng respetve quotes: If an amount of work d W s degraded to eat of temperature, te nrease n entropy s dsrr = dw /. In te ourse of te proess (one yle of a eat engne) te quantty of work W s stored n te form of te potental energy of te wegt. If we allow te wegt to snk bak to ts orgnal level and f ts energy s smultaneously transformed nto eat troug frton, te eat of frton f = W beng added to te soure o (te old reservor), te entropy of te latter wll nrease by D S = / = W /. f o o rr 41

4 Appled Pyss Resear Vol. 7, No. 5; 2015 ese last two values, t sould be noted, are dental to tose of Clausus termodynams. e entropy anges for wat an be alled te fundamental or essental transformatons araterzng Clausus formulaton aven been annotated n able 2 vs-à-vs tose of urrent seond law termodynams. e obvous dfferene exstng between tese two sets evne our pous asserton tat Clausus seond law termodynams and urrent seond law termodynams are essentally dfferent from one anoter. e values for te two transformatons takng plae n a ersble eat engne, tose upon w Clausus bult s seond law, are found n urrent seond law termodynams nexplably transtng from - / and (- / ) + ( / ), to a value of zero. able 2. Clausus set of values for te transformatons are ere ontrasted wt tose santoned by urrent termodynams. Note tat n te latter wle all ersble transformatons take plae wt zero entropy anges, all tose of an rersble nature onvey postve entropy anges ransformaton Clausus work C [ ( ) W] -/ 0 [ W ( )] / 0 [ W ( )] rr / / [ ( ) ( )] (- / ) + ( / ) 0 [ ( ) ( )] (- / ) + ( / ) 0 [ ( ) ( )] (- / ) + ( / ) (- / ) + ( / ) rr It s mpossble to mss from able 2 te fat tat urrent seond law termodynams assgns a zero total entropy ange to all ersble transformatons and a postve one to tose rersble. In assurng a zero total entropy ange for any ersble proess and a postve total entropy ange for tose rersble, ts set of values makes t mpossble to fault te law of nreasng entropy, and n te proess safeguards from any doubt well, almost any doubt- te entrop evoluton of te unverse and ts eventual demse n te form of ts eat deat. e evdene to be provded n wat follows wll unvel te fat tat te suess of C n provdng a foolproof body of knowledge was only apparent, as te proess of gettng tere nvolved, as already mentoned, a trade of flaws. Here we fnd te termodynam verson of tat old tale of dggng one ole to over anoter. It s to te dentfaton of te new logal ole tat te followng argument s dreted to. 4. e Untenable Poston of Current Seond Law ermodynams o understand wy te assgnment of a zero entropy ange to te ersble transformaton of eat nto work s unaeptable, let us onsder te proesses depted n Fgure 5. In t proess (a) represents te ersble transfer of an amount of eat d between two bodes (eat reservors) of essentally te same temperature. Proess (b) represents, on ts part, an sotermal and ersble deal gas expanson n w, onomtant to te transfer of te amount of eat d between two bodes te eat reservor and te gas- of essentally te same temperature, we fnd te produton of an equvalent amount of work dw by te opposed expanson of te gas. e essental fat tat needs ere to be onsdered s tat aordng to urrent termodynam wsdom bot of tese proesses are, entropy-wse, ndstngusable of one anoter as bot of tem take plae wt a zero total entropy ange. On ts perspetve energy upgradng proess (b), troug w te dsorganzed energy form we all eat beomes te ordered energy we all work, s ndstngusable from a eat transfer proess. For te seond law of termodynams, as urrently understood, work and eat are n ersble proesses- energy forms of te same qualty. e onstrutve, organzng, and transformng powers subsumed by work evdent n all te strutures around us, n te ange n ondton t an produe n oter bodes, n ts apablty of restorng spent gradents and make spontaneous te non-spontaneous et., are under ts perspetve erased, negleted, gnored. Wat besdes eatng a older body and oolng a otter one an be aeved by a eat transfer proess su as (a)? Notng! e fat tat none of te pous tasks an be aeved wt eat tself.e., wtout t beng frst onverted nto work, sould be reason enoug to rejet te noton of eat and work beng entropally ndstngusable n ersble proesses. ose termodynamsts denyng a qualty gan or entropy ange n te neessarly ersble- transformaton of eat nto work mgt tnk dfferently f one ter omes were deprved of eletrty, tey ad to rely on fre as energy soure. In a stuaton lke ts tey are most lkely to agree tat ter qualty of lfe as radally dereased. 42

5 Appled Pyss Resear Vol. 7, No. 5; 2015 d d dw (a) (b) Fgure 5. Proess (a) represents te transfer of an amount of eat d between two bodes of essentally te same temperature. Here no work s outputted. Proess (b) represents, on ts part, te sotermal and ersble expanson of an deal gas were an amount of eat d s also transferred between two bodes te eat reservor and te deal gas- at essentally te same temperature. e dfferene beng tat ere te expanson of te gas manages to transform d nto an equvalent amount of work dw tat appears n an approprate meanal reservor. For urrent seond law termodynams tese two proesses are entropally ndstngusable as tey bot take plae wt a zero total entropy ange. s poston makes eat and work energy forms of te same qualty In makng te wole new set of apabltes ganed n te transformaton of eat nto work takng plae n proess (b) ndstngusable from te pratally non-exstent apablty of te effets- barren eat transfer represented n (a), urrent termodynams s notng sort of equatng equlbrum araterzed by ts nablty for work output- wt evoluton, ange, or transformaton -te sne qua non ondton for work produton; or equvalently, of equatng rest wt moton. In ts last perspetve proesses (a) and (b) re-edt te onfrontaton between te plosopal postons of Parmendes and Heraltus. Wen onfronted wt t te orretors of Clausus work responsble for brngng forward C wasted no tme n fndng an effent soluton: ey smply, n a way of speakng, klled Heraltus! As te followng statements attest, te pous onsderatons are n lne wt te ntutve, ommonsensal lore exstng around te dfferent qualty of tese two energy forms: Entropy s a measure for te qualty of te energy n te system. (Jants, 1983, p. 25); In pratal terms wat a steam engne does s to onvert eat nto work, wt work smply meanng a more useful and organzed form of energy. (Coveney & Hgfeld, 1990, p. 149); But wat about ange of form of moton, or so-alled energy? If we ange eat nto meanal moton or ve versa, s not te qualty altered wle te quantty remans te same? (Engels, 1964, p. 64); tere an be no esape from te onluson tat tere s an ntrns dfferene between eat and work. (Smt & Van Ness, 1965, p. 175); e entropy ange of te system plus surroundngs may be vewed as a quanttatve measure, or ndex, of te degradaton of energy as work to energy as eat, as a onsequene of rersble elements n te proess under study. (Weber & Messner, 1957, p. 168). In reognzng an ntrns dfferene n qualty between eat and work, te pous notons appear to ontradt urrent seond law termodynams poston of makng ersble eat/work nter-onversons onstant entropy proesses. For f ndeed eat and work are energy forms of dfferent qualty ten any transformaton of one nto te oter as to be aompaned by a measure a non-zero entropy ange- of te qualty ganed or lost n t. Beyond determnng weter te ange of qualty s permanent or not, te nature of te pat (ersble or rersble) as no oter bearng n ts matter; ertanly not te power of makng te dfferene n qualty between eat and work appear under ertan ondtons, su as te rersble transformaton of work nto eat, and dsappear n oters, su as te ersble eat-to-work nter-onversons. As Clausus orgnally stated, te value of a ange from work nto eat must be proportonal to te amount of eat generated and beyond ts t an only depend on te temperature. (Clausus, p. 98). If so, te ersble transformaton of eat nto work must reflet te gan n qualty wt n aord to Clausus onventon on ts matter- an entropy derease, wle bot, te ersble and rersble transformatons of work nto eat must onvey postve entropy anges as a refleton of te transtory n te former and permanent n te latter- loss of qualty nvolved. Our agreement wt Clausus n tat te ersble nter-onversons between eat and work onvey entropy anges of magntudes 43

6 Appled Pyss Resear Vol. 7, No. 5; 2015 and D S [ ( ) W] = - / (54) D SW [ ( )] = / (55) mples neessarly our rejeton of urrent seond law termodynams. Even so, Equatons (54) and (55) are as far as our agreement wt Clausus work atually goes. e reason bend ts statement wll beome evdent n te followng setons. Let us ten make lear tat gven our rejeton of C wat we propose to do s go bak to Clausus work and guded - n addton to te notons represented by te last two equatons- by te lgt of reason and onstraned by logal soundness but ertanly not by preoneved notons or relgous or plosopal bas- subjet Clausus work to a profound ew. e body of knowledge to emerge from su an undertakng s wat onsttutes te Negentrop Formulaton (NF), n ts autor s opnon te rue Seond Law of ermodynams. Let us ten go bak to Clausus work n order to reognze s only logal flaw, te one from w all te nonsstenes unveled n Part I orgnate. 5. e Negentrop Formulaton 5.1 Clausus Coneptual Error or te rue Role of Isentrop Proesses n Carnot s Cyle Let us enter our attenton on te onatenaton of proesses EB-BC-CD-DA depted n Fgure 6(a), below. As sould be remembered t s ts onatenaton te one responsble for produng n a eat engne te ersble transfer of eat from te ot to te old reservor. As known, n sotermal and ersble expanson EB an amount of eat of temperature transferred from te ot reservor to te deal gas s transformed nto an equvalent amount of work W. In dong so proess EB brngs forward transformaton [ ( ) W] wt an assoated entropy ange, n agreement wt Equaton (54), of - /. If we assume a value of zero for te entropy of te unverse at pont E ten ts entropy at pont B s equal to SB =- /. Let us now reognze tat proess BC not only anges te volume and temperature of te gas from ts orgnal values at B to tose rulng at C; t also anges te eat reservor n ontat wt te gas from tat of temperature to tat of temperature. e meanal reservor and te work t safe-keeps form now part of te new stuaton. e fat tat te ondton of all tese bodes defnes at any moment te ondton of te unverse allows us to understand tat wat proess BC atually aomplses s te substtuton of unverse B wt unverse C, and n dong so brngs forward an entropy ange onsstng n swtng te negentropy ( - / ) assoated to te produton of W out of eat of temperature for tat w n te amount of - / would ave arsen ad W been generated from eat of temperature. e ratonale s atually very smple. One W and te rest of te unverse arrve at C, ts work beomes ndstngusable from a smlar amount generated out of eat of temperature. In te unverse exange produed by proess BC, W s n a manner of speakng deoupled from eat of temperature and oupled to eat of temperature. e palpable effet of ts deouplng /ouplng s te orreton from one temperature to anoter of te negentropy assoated to W. In dong ts, adabat and ersble proess BC brngs forwards wat n rgor s te transformaton of a transformaton,.e. {[ ( ) W] [ ( ) W]}, wt an assoated entropy ange (fnal entropy mnus ntal entropy) of (- / ) -(- / ) or equvalently, of (-W / ) -(- W / ). e deouplng / ouplng referred an also take plae, as wll be sown below, wt work remanng one an sotermal and ersble ompresson as taken plae. For eonomy reasons ts transformaton wll be from now on desgnated as te W-ransformaton and represented, n general, as [ W ( 1) W ( 2)] or as [ W ] BC. Wt ts notaton we an wrte D SW [ ( ) W( )] = D SW [ ] = (- / ) -(- / ) (56) BC Wt tese anteedents n plae we an wrte te entropy of te unverse at C as follows S = S +D S[ W] +D S[ BC] (57) B BC e pously gven entropy values for S B and [ W ] BC n ombnaton wt te fat tat proess BC s sentrop, allows wrtng te pous equaton as follows: S = (- / ) + [(- / ) -(- / )] + 0 = - / (58) e entropy ange assoated to wat we ave alled te W-ransformaton was unreognzed by Clausus. In s analyss proesses BC and DA are smple onnetors between te ot and old soterms. Had e been aware of t, te sene of termodynams would ave taken ts true form from te very begnnng. e effet n queston was also unreognzed by te orretors of Clausus work. Its oversgt ad, owever, no onsequene as te W-ransformaton n ts ase would be swtng a zero for a zero, as for C te entropy ange for te transformaton of eat nto work s zero at any temperature. 44

7 Appled Pyss Resear Vol. 7, No. 5; 2015 A D E B C W W A D B C W W (a) ΔS [ EB] = / ΔS [ BC] = ( / ) + ( / ) ΔS [ CD] = / ΔS[ DA] = 0 ΔS[ EBCDA] = 0 (b) ΔS [ AB] = / ΔS [ BC] = ( / ) + ( / ) Δ S [ CD] = / ΔS [ DA] = ( / ) + ( / ) ΔS [ ABCDA] = / Fgure 6. Fgure (a) represents te onatenaton of proesses requred to produe n a eat engne te ersble transformaton of eat from one temperature to anoter. If traveled n te sequene EBCDA te eat wll be ersbly transformed from te ot to te old reservor. If nstead te pat ADCBE s followed, eat wll be transferred n te opposte dreton. Fgure (b) represents, on ts part, a ersble ylal proess. e entropy anges wrtten below tese fgures are te one alulated aordng to te Negentrop Formulaton wt te onourse of ts assoated onept: e W-ransformaton Contnung wt te onatenaton of proesses beng onsdered, let us agree tat te unverse fnds tself now at C wt an entropy, as sown by Equaton (58), of - /. e next proess of te onatenaton s sotermal and ersble ompresson CD. Here te work W avalable n te meanal reservor s expended along ts proess and te spent work released as eat at te temperature of te old reservor. In dong so proess CD brngs forward transformaton [ W ( )] wt an assoated entropy ange of /. e entropy of te unverse at D an now be alulated as te summaton of te entropy at C plus te entropy ange of proess CD. Wen ts s done we get: SD = (- / ) + ( / ) = 0. s s a general result of te Negentrop Formulaton: For any two sotermal and ersble deal gas expansons EB( 1) and DC( 2 ) for w WEB = WDC (same g -1 g -1 amount of work produed by one and te oter) and V 1 B = V 2 C (ponts B and C lyng on te same ersble adabat), t s true tat SE = SD (e entropy of te unverse at B and D are dental). Let us now agree tat at pont D no more work s avalable n te unverse. Proess DA, te last leg of te onatenaton, as tus no work to deouple / ouple and onsequently no negentropy to orret from one temperature to anoter. Here proess DA assumes te role of smple onnetor between te ot and old soterm. If so: D S[0] = [(-0/ )-(- 0/ )] = 0 (59) erefore DA S = S +D S[0] +D S[ DA] = 0 (60) A D DA We are now n possesson of all te elements requred to alulate te entropy ange assgned by te Negentrop Formulaton (NF) to onatenaton EB-BC-CD-DA, or equvalently, to te ersble transformaton of from to. Reognton of te fat tat te sad onatenaton s defned by te transt of te unverse from ondton E, wt S E = 0, to ondton A, wt S A = 0, allows us to wrte D S [ ( ) ( )] = S - S = 0. An alternatve pat to ts result an be obtaned by wrtng NF A E D S [ ( ) ( )] = D S[ ( ) W ] +D S[ W ] +D S[ W ( )] +D S[0] (61) NF BC DA On substtuton of te orrespondng values for te entropy anges of te transformatons tere nvolved, te pous equaton beomes D S [ ( ) ( )] = (- / ) + [(- / )-(- / )] + ( / ) + 0 (62) NF 45

8 Appled Pyss Resear Vol. 7, No. 5; 2015 On performane of te ndated operatons ts equaton redues to If so, ten D S [ ( ) ( )] = 0 (63) NF D S [ ( ) ( )] = 0 (64) NF A omparson between Equatons (27) and (61) makes lear te pont of separaton between Clausus seond law and te Negentrop Formulaton. us, wle n te former te ersble transfer of eat takng plae n a ersble ylal proess omes out of te ombnaton [ ( ) W ] + [ W ( )] ; n te latter t omes out as te produt of te ombnaton of tose two same transformatons plus te two W-ransformatons tere operatng. s oneptual dfferene fnds expresson n te dfferent entropy anges ts transformaton onveys n one ase and te oter: postve n te former, zero n te latter. Let us also note ere tat even f for te wrong reason, C also assgns a zero entropy ange to ts transformaton. As Equaton (48) makes evdent, ts transformaton s n C ratonalzed te same way Clausus dd. e zero arses ere, owever, as a onsequene of C makng zero te entropy ange of eat-work nter-onversons. e rgt result appears ts way at te ost of ntrodung te noton assertng te qualty dentty between eat and work. s s wat we meant above wen we sad tat C removed one flaw at te ost of ntrodung anoter In order to fully araterze te Negentrop Formulaton and tus be able to ontrast t wt bot Clausus onstruton and urrent seond law termodynams we need te values for transformatons D S [ ( ) ( )] rr and D SW [ ( )] rr. For tese we wll adopt te values gven assgned to tem by C as gven n Equatons (52) and (53). e reason for ts bols down to te fat tat te proedure to obtan tem t s te same n C and NF. In te former, a dret applaton of te entropy defnng equaton to bot of te reservors nvolved n su a proess. In te latter we smply reognze te soundness of Smdt s reasonng gven n regard to Equaton (53). At ts pont te fundamental set of values of te Negentrop Formulaton s omplete. In order to smplfy te omparson between te values tat tese tree dfferent formulatons of te seond law of termodynams assgn to te ersble and rersble nter-onversons between eat and work as well as tose of eat from one temperature to anoter, s tat able 3 as been onstruted. Wt te Negentrop Formulaton properly araterzed va te set of orrespondng values of able 3, we an address te problem of determnng te total or unverse entropy ange by t assoated to a ersble ylal proess. For te same reasons advaned n regard to Equaton (51), te total or unverse entropy ange for te ersble yle wll be wrtten as te summaton of te entropy anges for te two transformatons tere takng plae,.e. D SNF [ Cyle] = D S[ ( ) W ] +D S[ ( ) ( )] (65) Replaement of te ndated entropy anges for te values annotated n able 3 produes D S [ Cyle] = - / (66) NF e pous result ndates tat ontrary to ommon wsdom, te total entropy ange assoated to one yle n te operaton of a ersble eat engne s negatve, n oter words t s a negentrop proess; te negentropy reated beng proportonal to te work produed. It needs to be reognzed ere tat at te end of one yle te only body returnng to ts ntal ondton s te deal gas atng as varable body. e oter tree bodes tere nvolved: te eat and meanal reservors, are found n a ondton dfferent tan te one tey orgnally ad. ese tree anges remanng n te unverse are (1) te one sustaned by te ot reservor n releasng an amount of eat ; (2) te one sustaned by te old reservor n reevng an amount of eat ; and (3) te one sustaned by te meanal reservor n reevng an amount of work W produed out of eat of temperature. e entropy anges of te eat reservors do not appear on Equaton (66) on reason of ter magntudes ombnng to a value of zero,.e. (- / ) + ( / ) = 0. e fat tat t s te produton of W te only non-ompensated ange left n te unverse explans wy te only term appearng n ts equaton s - /. If to Equaton (66) we now add a zero n te form of (- / ) + ( / ) we wll be gettng wat n all rgor s te Negentrop Formulaton s expresson for te total entropy ange of a ersble yle: D SNF [ Cyle] = (67) Gven te radally new perspetve brougt about by te W-ransformaton, an analyss smlar to tat leadng to Equaton (66) wll be arred on for te ersble yle ABCDA depted n Fgure 6(b). Let us start by wrtng te entropy anges assoated to proesses AB, [ W ] BC, CD, and [ W ] DA, as follows 46

9 Appled Pyss Resear Vol. 7, No. 5; 2015 D S [ AB] = D S [ ( ) W ] = - / (68) NF NF D S [ W ] = (- / )-(- / ) (69) NF BC D S [ CD] = D S [ W ( )] = / (70) NF NF D SW [ ] = (-/ ) -(- / ) (71) DA e pous equatons reflet te followng fats: (1) tat n te ase beng onsdered proess AB transforms an amount of eat ( ) nto an equvalent amount of work W wt an assoated entropy ange of - / ; (2) e W-ransformaton operatng along proess BC orrets te negentropy arred on by W as part of te ot soterm for tat t onveys as part of te old soterm; (3) e porton W out of te amount of work W avalable n te meanal reservor s used to arry on ompresson CD. e spent work ends up as an equvalent amount of eat n te old reservor. e entropy ange for ts proess s / ; (4) e amount of work avalable n te meanal reservor at pont D s W = W - W. As part of te old soterm, te negentropy assoated to W amounts to - /. e W-ransformaton operatng along DA orrets ts value for te one tat orresponds to W as part of te ot soterm. able 3. e set of values araterzng te Negentrop Formulaton s ere sown vs-à-vs tose of Clausus onstruton and urrent seond law termodynams. Note tat te dfferene wt Clausus work omes down to te entropy anges of te ersble transformatons of eat ransformaton Clausus C NF [ ( ) W] [ W ( )] 0 [ W ( )] rr [ ( ) ( )] 1 1 ( - ) 0 0 [ ( ) ( )] 1 1 ( - ) 0 0 [ ( ) ( )] rr 1 1 ( - ) 1 1 ( - ) 1 1 ( - ) Sne te ombnaton of proesses AB, [ W ] BC, CD, and [ W ] DA defne te ersble yle beng onsdered, te summaton of ter entropy anges must defne ts total (or unverse) entropy ange. erefore D S [ Cyle] = (- / ) + [(- / )-(- / )] + ( / ) + [(- / )-(- / )] (72) NF Performane of te ndated operatons produes te followng result D S [ Cyle] = - / (73) NF e agreement exstng between Equatons (66) and (73) s a testament of te self-onssteny of te Negentrop Formulaton. In te next seton t wll be sown tat Equaton (67) retans ts form wen appled to rersble ylal proesses. 5.2 e Negentrop Formulaton on Irersble Cylal Proesses e problem at and s to determne n aord wt te Negentrop Formulaton te total entropy ange for a non-ersble ylal proess. Let us ten fous our attenton on Fgure 7. ere proess (a) represents a ylal proess w workng wt an effeny, á, manages to produe an amount of work W = out of an amount of eat reeved by t from te ot reservor of temperature. In te fgure, proesses (b) and () represent te ersble and rersble omponents of (a). In proess (b), and represent, respetvely, te amounts of eat omng out of te ot reservor, and taken n by te old reservor ad te produton of W taken plae n a ersble fason. Here - = = W and W =. On 47

10 Appled Pyss Resear Vol. 7, No. 5; 2015 reason of ñ we ave tat W / ñ W / or, equvalently, tat we an wrte te total entropy ange for proess (b) as: ñ. In aord wt Equaton (66) D S = - W / = - / (74) at te same work output demands a larger ntake of eat n (a) tan n (b) an only be explaned by te fat tat te amount of eat quantfyng te dfferene - =, manages to bypass n (a) te work produng rut of te varable body. In ts rersble transt ts amount of eat arres wt t an untapped work produng potental, te lost work, n te amount of W = ( - )/ (75) lost If ts work produng potental ad been arnessed ten nstead of W, proess 7(a) would ave ad a work output of W + Wlost = W =. W = W + ) A smple ylal proess (a (b) () e ersble omponent e rersble omponent ΔS NF [ Cyle] rr ΔS ΔSrr Fgure 7. A smple ylal proess of effeny s ere separated nto ts ersble and rersble omponents. e entropy ange for ts proess s determned by te summaton of te ontrbutons of ts omponents e rersble omponent depted n () refers to ts amount of eat flowng n a dret, rersble fason from te ot to te old reservor. In orrespondene to ts rersble nature ts total entropy ange amounts to D S = ( - )/( ) = W / (76) rr lost Let us also note tat on reason of - = -, we an wrte - = - =. Implt n ts last equaton s te fat tat ersble s any yle for w = 0. Let us note tat ombned, proesses (b) and () reprodue proess (a). us, + =, + =, and Wb ( ) + W ( ) = Wa ( ). It s on reason of ts tat we an express te total entropy ange of proess (a) as te summaton of te total entropy anges of ts ersble and rersble omponents (b) and (), as D S [ Cyle] = D S +D S (77) NF rr rr In order to be able to wrte te fnal expresson for D SNF[ Cyle] rr we wll proeed to perform some requred algebra manpulatons on D Srr as gven by Equaton (76). Let us start by wrtng Equaton (76) as follows D S = - rr + (78) e respetve replaement of by - and - n te frst and seond terms on te pous equaton leads, after rearrangement, to te followng expresson D Srr = (- + )-(- + ) (79) 48

11 Appled Pyss Resear Vol. 7, No. 5; 2015 Reognton of te fat tat te last of te rgt and sde parentess s equal to zero redues Equaton (79) to te followng form. D S = - rr + (80) Equaton (80), t sould be noted, s wat C onsders to be te total or unverse entropy ange for proess (a),.e. D SC [ Cyle] rr = D S +D Srr = 0+D Srr = D Srr (81) e fat tat D Srr s notng more tan te re-expresson of Equaton (78),.e. te re-expresson of te total entropy ange for te rersble transfer of, allows us to relate te pous equaton to te lost work assoated to ts eat transfer,.e. Wlost D SC [ Cyle] rr = D Srr = [ ( - )/ ]/ = = - + (82) It s troug ts onneton tat we an get a glmpse at te essental message of C. Lookng at te pous equaton as well as to te representaton of te proess to w t refers proess 7()- we an understand tat urrent seond law termodynams onsders te engne of nature as avng one and only one apablty: te wasteful dsspaton of gradents.e. te produton of lost work. As an be seen n Equaton (81), alen to urrent seond law termodynams entrop balane of te unverse s te negentrop ontrbuton assoated to te upgradng of eat nto work. All te onstrutve, orderng, and organzng apabltes of work ave been erased from termodynams by te urrent formulaton. No better explanaton of ts poston an be offered tan te desre to retan at all osts te law of nreasng entropy as te supreme law of te unverse. Wt unquestonable fat on te dtum uod non est n formula non est n mondo (von Bertalanffy, 1975, p. 70) te orretors of Clausus work smply gnored te negentrop ontrbuton of transformaton [ ( ) W] and n dong so restrted te doman of applaton of urrent seond law termodynams to rersble penomena. e pereved, atually fatual mpotene of C for advanng reasonable, logally sound, and testable models for self-organzng penomena fnds ts ause and explanaton n te fat tat n negatng te onstrutve role of work, C took away te only analytal tool apable of makng sense of tese penomena. e evdent ontradton exstng between te message of te law of nreasng entropy and te order-produng apablty of te unverse evdent wtn and around us as been suntly expressed by Callos as Clausus and Darwn annot bot be rgt (1973). At te eyes of te teores of self-organzaton fndng sustenane n urrent termodynam wsdom, ts nessant emergene of order and organzaton are notng more tan te blnd atons of a self-destrutng and stupd unverse eager to dssolve tself n te Götterdämmerung of ts eat deat, as aordng to tese teores every self-organzed struture takes form and develops at te pre of nreasng te rate of entropy produton (Prgogne, 1980, p. 89) For te Negentrop Formulaton te term D Srr, as gven by equaton (80), s notng more tan te rersble omponent of te total entropy ange. For ts formulaton, te total or unverse entropy ange s determned, as sown by Equaton (77), by bot, te ersble and rersble ontrbutons. e fat tat te ersble ontrbuton omng out of proess (b) amounts to D S = - W / (83) Allows us to wrte as follows te orret expresson for te total entropy ange of proess (a) D SNF [ Cyle] rr = (84) Evdent n Equaton (67) and (84) s te fat tat n te NF an equaton of te same form apples to ersble and rersble ylal proesses alke. 5.3 Irersble Proesses from te Opposte s Perspetve Let us now agree tat as sown n Fgure 8, te operaton of proess (a) s bound by te rersble and ersble operatons sown tere, respetvely, as (d) and (e). s an be understood by realzng tat for te gven ondtons of operaton represented by and te temperatures of te reservors, te most effent operaton possble s te ersble operaton sown n (e). s operaton and ts total entropy ange - W / represent te effent lmt te lmt along a pat of nreased effenes- for proess (a) and for ts total entropy ange. Lkewse, te realzaton tat represents te largest amount of eat tat an be rersbly transferred from te ot to te old reservor makes of proess (d) te most neffent operaton possble. s 49

12 Appled Pyss Resear Vol. 7, No. 5; 2015 operaton and ts total entropy ange ( - )/( ) represent te neffent lmt te lmt along a pat of dereasng effenes- for proess (a) and for ts total entropy ange. ese lmtng operatons wll also be referred to as te ersble and rersble oppostes and respetvely symbolzed as [ REV ] and [ IRR ]. As t s known, n ersble proess (e), were W = W, te wole of te unverse s suseptble of beng restored to ts prese ntal ondton va te smple expedent of feedng W to te ersble nverse of (e). For proess (d), were W = 0, no porton of te unverse s suseptble of restoraton wtout te ourrene of addtonal anges. Here any restoraton wll ome at te pre of anges beng left n tat body alled to supply te work requred to propel te refrgerator wt w te restoraton s to be arred troug. At te lgt of te pous onsderatons t souldn t be dffult to understand tat n proess (a), were 0 áw á W, only a porton of te unverse an be restored wtout addtonal anges settng n. It s n ts regard tat any su proess s partally ersble. s an be understood f t s realzed tat by feedng te work outputted by proess (a) to a ersble refrgerator t s possble to restore to ts ntal ondton tat porton of te unverse nvolved n te produton of te sad work, wtout te ourrene of any oter permanent ange. Larger outputs of work lead to larger fratons of te unverse suseptble of restoraton, and ve versa. Sne te wole of te unverse s suseptble of beng restored wen W = W, wle none s wen W = 0, t follows tat te restorable porton.e. te ersble fraton of te orgnal unverse, wll be gven by W / W. s quotent wll be subsequently referred to as te ersblty degree, and symbolzed as φ,.e. φ = W /W (85) Obvously, te lmt values for φ are, respetvely, zero and one for te rersble and ersble lmts pously desrbed. W W, (d) (a) (e) e rersble A smple ylal e ersble opposte proess opposte η = 0 η Fgure 8. e ersble and rersble oppostes to ylal proess (a) are ere respetvely sown n fgures (e) and (d) η It sould be noted tat an equvalent defnton to te one just gven an be wrtten for φ as te quotent of te effeny of te ylal proess (a) and tat of ts ersble lmt, as follows: φ = η / η (86) s alternate defnton seds lgt on te fat tat φ s ultmately a normalzed or redued knd of effeny. Aordng to te pous onsderatons, te total entropy ange for proess (a) wll be lmted on te neffent, rersble sde by te total entropy ange of proess (d) gven below ΔS NF [ IRR] = ΔS NF [ φ = 0] = ( ) /( ) 0 (87) And on te opposte sde, te effent one, te lmt wll be te negatve total entropy ange of te ersble lmt: W ΔS NF [ REV ] = ΔS NF [ φ = 1] = 0 (88) 50

13 Appled Pyss Resear Vol. 7, No. 5; 2015 e atual ontrbutons made by tese lmtng operatons to te total entropy ange of ylal proess (a) depend on te relatve wegt ea of tese extremes as n te atual proess. e ersble ontrbuton, determned by te amount of work W atually produed by ts ylal proess, wll be gven by - W /, w an also be wrtten as ( W / W )(- W / ), or as φ ΔS NF [REV ]. e eat beng rersbly transferred from te ot to te old reservor wll be quantfed as follows: at an amount of work W s produed, wt W á W, means tat of all te eat gven off by te ot reservor only te porton ( / W) W s used for te produton of W. e rest: - ( / W) W or ( 1 φ) s onsequently tat beng rersbly transferred; and to ts amount orresponds an entropy ange of ( 1 φ ) ( ) /( ), w an also be wrtten as ( 1 φ ) ΔS[ IRR]. At te lgt of te results just gven te total entropy ange for proess (a) an be wrtten as follows: ΔS NF [ Cyle] rr = (1 φ ) ΔS[ IRR] + φ ΔS[ REV ] (89) Or as: W - W D SNF [ Cyle] rr = (1- ) ( )- (90) W e fats tat ( - )/ = W and W - W = ( -, )-( - ) = -, allows us to wrte Equaton (90) as follows e addton and subtraton of / to Equaton (91) leads us to W, D SNF [ Cyle] rr = - - (91) W D S Cyle = (92), NF[ ] rr ( ) Reognton of te fat tat te mddle term of te pous equaton s equal to zero allows us to wrte te Negentrop Formulaton s expresson for te total entropy ange of proess (a) as follows W D SNF [ Cyle] rr = = (93) Beyond te realzaton tat ts alternatve desrpton of ylal proess (a) leads to te same equaton for te total entropy ange of an rersble ylal proess as tat obtaned n te pous seton n referene to Fgure 7, t also, and peraps more mportant, allows us to realze tat te total entropy ange of any su proess s determned by te wegted ontrbutons of te operatons rulng at te opposte ends of te effeny or ersblty degree doman [0,1]. On ts perspetve te partular entropy ange a gven ylal operaton may produe wll be te result of te partular balane tese oppostes adopt n te stuaton beng onsdered. Low values of φ mply te palene of te rersble opposte and te predomnane of ts entrop ontrbuton over te negentrop ontrbuton of te ersble opposte. Hg values of φ mply, on te oter and, te palene of te ersble opposte and te predomnane of ts negentrop ontrbuton over te entrop ontrbuton of te rersble opposte. e fat tat aordng to ts pture te total entropy ange of a ylal proess transts from a postve to a negatve value as φ transts from zero to one, and ve versa, means tat tere must exst an operaton ntermedate between te two extremes for w D SNF [ Cyle] rr = 0. Substtuton of ts ondton n Equaton (93) followed by adequately solvng for /, leads to te followng expresson: - / = + Replaement of te quotent /, te effeny of te operaton at te gven ondton, by umbral = /, allows us to wrte Equaton (94) as follows umbral = e name gven to ts effeny omes from te fat tat t separates te entrop operatons of proess (a) from tose negentrop, or better sad te entrop from te negentrop unverses. From our pous defnton of φ t follows tat φ umbral = ηumbral / η, and f so - + (94) (95) 51

14 Appled Pyss Resear Vol. 7, No. 5; 2015 f umbral = + In terms of tese varables we an say tat any operaton for w φ φumbral (or η ηumbral ) wll take plae wt a total postve entropy ange; wle any one for w φ φumbral (or η ηumbral ) wt a negatve total entropy ange. e partular operaton for w φ = φumbral s, on te oter and, non-entrop. e reader sould note tat proess 7() s equvalent to te (1- ϕ) fraton of proess 8(d), and proess 7(b) to te ϕ fraton of proess 8(e) Approprate ombnaton of Equatons (77) and (81) allows us to wrte te followng expresson relatng te total entropy anges assgned by NF and C to a Carnot s yle (96) D S [ Cyle] = DS [ Cyle] -( W / ) (97) NF C From t we learn tat te negentrop formulaton subsumes urrent seond law termodynams as tat speal ase araterzed by W = 0, or equvalently, by = 0. In oter words t s at te rersble lmt tat te negentrop formulaton ollapses, redues to C. 5.4 Bak to te Law of Inreasng Entropy From our pous dsussons we ave learned tat n NF, rersble proesses an, dependng on ter effeny, take plae wt postve, negatve, or zero total entropy anges, result tat denes te noton gvng sustenane to te law of nreasng entropy, namely, tat all rersble proesses take plae wt postve total entropy anges. s noton s notng new. It atually orgnates n Clausus formulaton. o prove ts let us reall tat n Seton 2.12 t was proved, n regard to te rersble proess depted n Fgure 4, tat for = 600 K, = 400 K, and = 1200 J ts total entropy ange, as expressed n Equaton (36), amounted to J K -1, a proof tat n Clausus formulaton rersble proesses wt negatve entropy anges are possble. If we repeat tese alulatons for te rersble proess takng plae wt = 1200 J, = 600 K and = 300 K we wll get a total entropy ange n te amount of zero. A fnal alulaton wt = 1200 J, 1 = 600 K and = 298 K produes for ts rersble proess a total entropy ange of 0.1J K -. e pous argument leads to te nevtable onluson tat te law of nreasng entropy, n beng nompatble wt Clausus formulaton, was dead ab ovo. How ten an Clausus endorsement of su a law be explaned? ese onsderatons, n ombnaton wt te unexplaned sft n values araterzng te transt from Clausus work to C evdent n able 2, makes te valdty dsplayed by ts law n urrent seond law termodynams as f orgnatng n wllful desgn, n deree. (See Note 2) 5.5 e Negentrop Formulaton and Its estable Predton e oneptual foundaton of te negentrop formulaton of te seond law of termodynams s onsttuted by te non-zero entropy anges assgned n able 3 to te ersble nter-onversons between eat and work, as well as by ts assoated onept te W-ransformaton. e applaton of tese two notons to te ersble and rersble operatons of Carnot s engne as unveled two essental fats: (1) tat te total entropy ange of tese proesses s determned by te ϕ-wegted, opposte sgns ontrbutons of te ersble and rersble oppostes onsttutng ter effent and neffent lmts, and (2) te exstene of a partular operaton te umbral operaton- n w te ombnaton of te sad opposte sgn ontrbutons, n avng te same magntude, produe a non entrop unverse. e relevane of tese results onssts n te realzaton tat any proess workng wt effeny smaller tan te umbral s entrop, wle tose wt effenes larger tan te umbral, negentrop. s knowledge s mportant beause t supples a termodynam model for te emergene of order n self-organzng penomena. However logal or ommonsensal tese results appear to be, te fat s tat at ts pont te negentrop formulaton s notng more tan te rgorous elaboraton of te onjeture assgnng non-zero entropy anges to transformatons [ ( ) W] and [ W ( )]. In order to beome a sentf onstruton eter ts bas notons or te results from tey obtaned must ome under te judgment of experene. Let us ten propose te followng noton as a testable result of te Negentrop Formulaton: e emergene of order n self-organzng penomena s te tangble, observable manfestaton of te transton of te unverse of tese proesses from an entrop to a negentrop ondton. In ts perspetve te so alled rss pont at w organzaton emerges plays a role omparable to te role played by te umbral effeny n eat engnes separatng entrop from negentrop operatons. 52

15 Appled Pyss Resear Vol. 7, No. 5; 2015 It needs to be mentoned ere tat even f te present work s not te frst one tat ts autor produe wt te non-valdty of bot, Clausus and urrent seond law termodynams as entral teme, t s ertanly te last. As su t supersedes all ts anteedents, among tem (Íñguez, 1999, 2011, and 2014) 5.6 A Note on omson s Dtum about te Restoraton of Meanal Energy In an 1852 paper Lord Kelvn delared Any restoraton of meanal energy, wtout more tan equvalent of dsspaton, s mpossble n nanmate materal proesses, and s probably never effeted by means of organzed matter, eter endowed wt vegetable lfe or subjeted to te wll of an anmated reature. (omson, 1852) In order to understand omson s dtum as well as to dentfy te oneptual error bend t let us refer agan to Fgure 7. As already explaned, proess 7(a) s a ylal proess of effeny. Its ersble and rersble omponents are tose sown n Fgures 7(b) and 7(), respetvely. As we know, any ersble proess produes te maxmum possble amount of work at ts partular ondtons of operaton. s means tat no work s lost n su a proess, or n oter words, tat te eat dsarded to te old reservor by any ersble proess arres wt t no unused or wasted work produng potental. Any lam n te ontrary mples te possblty of effenes larger tan te ersble effeny. For partally ersble proesses te stuaton s dfferent. Here tere s always an amount of eat reang te old reservor avng pously bypassed te work produng rut of te varable body. e amount of work ts eat ould ave produed, but ddn t, represent te wasted work produng potental, te lost work. Let us now assume tat one a yle n te operaton of 7(a) as taken plae, we are to use W to restore to te ot reservor all of te eat dsarded to te old reservor. A lttle ponderng about te ntended task would onvne us tat t s mpossble. e most W an do, as te nverse of proess 7(b) ndates, s to restore to te ot reservor. One ts transfer bak as been ompleted, tere s stll te amount of eat to be transferred but no work left to arry t on. Any attempt to do so requres work to be mported from somewere else. Under ts seme of tngs te restoraton of not only onsumes te work generated n 7(a); t also onsumes an extra amount of work for te restoraton of. At te lgt of tese results omson appears to be orret. Any restoraton of meanal energy wtout furter dsspaton appears mpossble. If te orgnal proess ad been ersble, would ave been zero. In ts stuaton te work produed by 7(a) would ave been presely te amount requred to brng bak to te ot reservor. It s only under ersble ondtons tat on omson s perspetve we an break even. s senaro, even f reasonable at frst sgt, s, owever, norret. o understand ts we ave to reall tat te amount of eat tat n Fgure 7 we ave represented as subsumes bot: and. e fat tat te transfer of s ersble means, as pously noted, tat no wasted work produng potental or lost work s arred by t to te old reservor. If ts s so, ten, as a matter of elementary log, we would ave to onlude tat f notng s lost wt ten tere s notng from t to be reuperated. If any reuperaton, any restoraton s to be attempted, t wll ave to be done n regard to and te unused work produng potental t arres wt t n ts rersble flow. e entropy ange assoated to ts rersble transformaton s ndatng us tat ts eat ould ave delvered an amount of work ( - )/. e fat tat ts s te only meanal energy lost makes t also te only one tat an be reuperated. After all we an t reuperate wat we aven t lost, an we? At ts pont te followng queston appears natural: at wat effeny sould proess 7(a) as to operate n order to produe te prese amount of work requred for te restoraton of troug a ersble refrgerator? In order to answer ts queston let us start wrtng te followng equaton for te amount of work requred to be outputted by proess 7(a) to aompls ts task: W = ( - )/. Substtuton ere of = - followed by te multplaton of te resultng expresson by ( / )( / ) leads, after some algebra and furter use of te expresson W = ( - )/, to te followng result: W = ( - )/( + ), or equvalently, to te realzaton tat workng at te umbral effeny proess 7(a) outputs te amount of work requred for te restoraton of to te ot reservor. Implt n tese results s te noton tat n any proess workng at ñ umbral tere wll be a work surplus after restoraton s aomplsed. s s te orret perspetve to omson s dtum (Íñguez 2012, pp ). 5.7 Negentropy as te Arrow of me Aordng to Casson (1998, p. 17) Wen matter and radaton were stll equlbrated n te Radaton Era, only a sngle temperature s needed to desrbe te termal story of te unverse However, one te Matter Era began, matter beame atom, te gas-energy equlbrum was destroyed, and a sngle temperature was nsuffent to spefy te bulk evoluton of te osmos. As tngs turned out, sne te random motons of te ydrogen and elum atoms faled to keep pae wt te general expanson of te atoms away from one anoter te matter ooled faster tan te radaton Su a termal gradent s te patent sgnature of a eat engne, and t s ts ever-wdenng gradent tat enabled matter to buld tngs 53